Finite Group DatabasebetaFinite Group Databasebeta| order | 320 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| Sylow | C5 ⋊ #557 |
| subgroup | G/H | |
|---|---|---|
| center, Frattini | C22 | C23×D5 |
| commutator, socle, prime radical, cosocle | C22×C5 | C24 |
| Fitting | C4×C5×Q8 | C2 |
| hypercenter | C4×Q8 | D5 |
| hypocenter | C5 | #557 |
| LCS | G ▷ C22×C5 ▷ C5 | |
| UCS | C1 ◁ C22 ◁ C4×Q8 | |
200 conjugacy classes of subgroups.
| minimal degree | μ(G) = 21 |
| transitive degree | μt(G) = 320 |
A minimal faithful permutation representation — 4 generators on 21 points:
(1,2,4,5,3)(6,8,9,13)(7,10,11,12)(14,17,18,16)(15,19,20,21);(14,16,18,17)(15,19,20,21);(14,15,18,20)(16,21,17,19);(2,3)(4,5)(6,7,9,11)(8,12,13,10)Polycyclic presentation:
2^6,5|a^2=e,b^2=f,c^2=f,d^2=ef,d^a=de,g^a=g^4,c^b=cf65 conjugacy classes — view the full character table.
| maximal subgroup of | #10611355, #10611358, #10611364, … |
| maximal normal subgroup of | #10611355, #10611358, #10611364, … |
| maximal quotient of | #10605373, #10605383, #10605389, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(320, 1241);